Problem: Let N be a positive odd number.

There are N coins, numbered 1, 2, \ldots, N. For each i (1 \leq i \leq N), when Coin i is tossed, it comes up heads with probability p_i and tails with probability 1 - p_i.

Taro has tossed all the N coins. Find the probability of having more heads than tails.

Constraints

* N is an odd number.
* 1 \leq N \leq 2999
* p_i is a real number and has two decimal places.
* 0 < p_i < 1

Input

Input is given from Standard Input in the following format:


N
p_1 p_2 \ldots p_N


Output

Print the probability of having more heads than tails. The output is considered correct when the absolute error is not greater than 10^{-9}.

Examples

Input

3
0.30 0.60 0.80


Output

0.612


Input

1
0.50


Output

0.5


Input

5
0.42 0.01 0.42 0.99 0.42


Output

0.3821815872

Solution (in JAVA):

Solution:
import java.util.*;

public class Main {

    static List<List<Integer>> edges = new ArrayList<>();
    static int[] dp = null;
    public static void main(String[] args) {
        Scanner sc = new Scanner(System.in);
        int N = sc.nextInt();
        double[] ps = new double[N];
        for (int i = 0; i < N ; i++) {
            ps[i] = sc.nextDouble();
        }

        double ans = 0;

        double[][] dp= new double[N+1][N+1];
        dp[0][0] =1;
        for (int i = 1; i <= N ; i++) {
            for (int j = 0; j <= i+1 && j<=N  ; j++) {
                if(j ==0 ){
                    dp[j][i]  =dp[j][i-1]* (1- ps[i-1]);
                }else{
                    dp[j][i]  =dp[j-1][i-1]* ps[i-1]
                                +dp[j][i-1]* (1- ps[i-1]);
                }
            }
        }

        for (int i = (N+1)/2; i <=N ; i++) {
            ans += dp[i][N];
        }

        System.out.println(ans);
    }

}